Well-posedness of the Hele-Shaw-Cahn-Hilliard system

Abstract

We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. For initial data in $H^s,s>\frac{d}{2}+1$, the existence and uniqueness of solution in $C([0,T];H^s)\cap L^2(0,T;H^{s+2})$ that is global in time in the two dimensional case ($d=2$) and local in time in the three dimensional case ($d=3$) are established. Several blow-up criterions in the three dimensional case are provided as well. One of the tools that we utilized is the Littlewood-Paley theory in order to establish certain key commutator estimates.

References in corpus (1)

• End-point Estimates and Multi-parameter Paraproducts on Higher Dimensional Tori